The Almagest Treatment of the Outer Planetsdduke/outer3.pdfinner planets Mercury and Venus, there is a substantial gap between Ptolemy’s description of events and what actually happened.3

Ptolemy’s Treatment of the Outer Planets

Dennis Duke, Florida State University

A primary theme of Ptolemy’s Almagest,1 virtually unique among known classical

sources, is that one can use empirical observations to determine numerical values for the

parameters of geometrical planetary models. The principal examples are:

1. an autumn equinox, a spring equinox, and a summer solstice, each of which is

combined with an ancient observation to determine the length of the tropical year.

Also, the resulting season lengths are used to determine the eccentricity and the

direction of apogee for the Sun.

2. a trio of lunar eclipses which determine the lunar parameters at syzygy. Virtually

identical lunar parameters result from Ptolemy’s reports of three earlier lunar

trios.

3. several lunar elongations are used to determine the parameters of the lunar model

away from syzygy.

4. two pairs of observations of Mercury that determine the longitude of apogee in

Ptolemy’s time, and six additional ancient observations that determine it some

400 years earlier.

5. two pairs of observations of Venus that determine the longitude of apogee in

Ptolemy’s time.

Dennis Duke Page 1 4/29/2004

Detailed analysis2 has shown that each of these sets of observations share a common,

conspicuous set of qualities. First, they are in most cases redundant. While using

redundant sets of data is in general a good idea, real concern arises in Ptolemy’s cases

when the redundant sets invariably result in essentially identical values for the parameters

being determined. Second, the parameters so determined are sometimes rather far from

their optimum values. Examples include the dates of the equinoxes and solstices, which

are all about a day late, and the direction of Mercury’s apogee, which is some 30° off.

Third, the deduced parameters are sometimes too close to their optimum values, even

though the quality of the underlying observations is too poor to expect such good

agreement. Altogether, we can safely conclude that at least for the Sun, the Moon and the

inner planets Mercury and Venus, there is a substantial gap between Ptolemy’s

description of events and what actually happened.3

The situation for the outer planets is, however, different. Ptolemy abandons his custom of

quoting redundant observations, and adopts instead a rather minimalist style, quoting

precisely the number of observations required to determine his parameters, and no more.

Given the Almagest bisected-equant model, exactly five observations are required. Three

mean oppositions determine the eccentricity, the longitude of apogee, and the mean

longitude at some time. A fourth observation4 determines the radius of the epicycle, and a

fifth observation, from about 400 years earlier, provides a high precision determination of

the mean motion in anomaly. The assumption that the radius of the epicycle always

points toward the mean Sun effectively allows determination of (a) the angle of the

epicycle with the epicycle apogee at some time, and (b) the mean motion in longitude


from the mean motion in anomaly and the length of the tropical year, thus completing the

determination of all the model parameter values needed to compute the position of the

planet at any moment in time. Ptolemy’s minimalist style extends to treating each planet

identically, to the extent of using virtually identical prose throughout the three

discussions of parameter derivations.

Suspicions that Ptolemy’s account might be less than candid are not new. Newton raised

the issue in light of (a) the failure of Ptolemy’s data to reproduce exactly the mean

motions in anomaly that Ptolemy uses in his tables, (b) the fact that for the fourth

observation for each planet, redundant determinations of the longitude with respect to a

star and the Moon agree exactly with each other, and (c) the fact that Ptolemy’s data for

Mars lead to almost exactly a round value for Mars’ eccentricity.5 Thurston pointed out

that the sequence of successively rounder values that Ptolemy finds for 2e for each planet

are not what one expects from an iterative solution, suggesting that Ptolemy was in fact

working backward from those round parameters.6

The purpose of this paper is to investigate whether for the outer planets Ptolemy followed

his otherwise consistent custom of describing a scenario that did not happen as he says, or

whether, at least for the outer planets, he left us a more accurate rendition of events. The

detailed reconstructions of Ptolemy’s calculations that follow show that, at least in the

Almagest, Ptolemy is a writer with consistent habits when it comes to observations. We

begin by reviewing, with minimal editorial comment, Ptolemy’s calculations for each

planet.


Mars

The input data for the Mars analysis are three oppositions with the mean Sun and two

additional observations, one from several hundred years before Ptolemy’s time. Ptolemy

reports times and the corresponding longitude of Mars at those times:

Ptolemy Model

t1 1768888.54167

t2 1770418.37500

t3 1771974.41667

t4 1771977.35903

t5 1622092.75000

λ1 81;00° 80;58,54°

λ2 148;50° 148;46,24°

λ3 242;34° 242;32,04°

λ4 241;36° 241;35,12°

λ5 212;15° 212;16,32°

λ1(S) 260;58,54°

λ2(S) 328;50,22°

λ3(S) 62;31,44°

Ptolemy typically gives the times to the nearest hour and does not correct for the equation

of time, which would be a negligible effect in the present context. I have converted his

times to standard astronomical Julian day numbers relative to noon at Alexandria. The


third column gives the longitudes predicted for Mars and the mean Sun using the final

Almagest models at Ptolemy’s specified times.

Ptolemy’s first task is to use three mean oppositions to determine the eccentricity and

longitude of apogee. The oppositions cannot, of course, be directly observed, and so must

be estimated from a set of data. Ptolemy refers to this twice. In the preface to the analysis

of the Mars trio of oppositions he writes7

Just as, in the case of the moon, we took the positions and the times of three

lunar eclipses and demonstrated geometrically the ratio of anomaly and the

position of the apogee, in the same we here too, using three acronychal

oppositions with respect to the mean course of the sun for each of these

planets, we observed as accurately as possible the positions by means of

astrolabic instruments [i.e. the armillary] and, on the basis of the mean courses

of the sun at the times of the observations, we further calculated the time and

position corresponding to greater precision in the diametrical opposition, we

demonstrate from these the ratio of eccentricity and the apogee.

Second, while describing the second and third oppositions for Saturn, which he says

occurred during daylight, he writes “We computed the time and place of exact opposition

from nearby observations…”. Using a zodiacal armillary the position of the planet could

be measured relative to a star and/or the Moon, a procedure Ptolemy claims to use for the

fourth observation of each outer planet, and for Mercury and Venus. The armillary also


gives the degree of the ecliptic culminating at the time of the measurement, and that can

be used to estimate the time of the observation. The underlying solar model gives the

longitude of the mean Sun at the same moment, and one then easily and accurately

estimates the moment of opposition by interpolation in the resulting table of data. The

fractional endings that Ptolemy reports for the longitudes of opposition: 00, 50, 34, 11,

54, 23, 13, 40, and 14 are consistent with the idea that the oppositions are computed using

some such process of data reduction, and not from direct analog measurement with an

armillary, which would certainly give only round fractions of a degree. Ptolemy’s

description of the measuring process does not, of course, necessarily imply that Ptolemy

himself used it to make any measurements (see the discussion in the Comments section,

below).

While a simple eccentric model can be solved directly, an iterative scheme is used to

solve the bisected equant model (see Appendix A). The actual input data to Ptolemy’s

numerical analysis are, in the notation of the Appendix, the angles α, β, γ, and δ.

Ptolemy’s values compare to exactly computed values, using Ptolemy’s times and

longitudes, as follows:

Ptolemy Exact

α 81;44° 81;43,26°

β 95;28° 95;27,31°

γ 67;50° 67;50°

δ 93;44° 93;44°


In the notation of the Appendix, Ptolemy gives8 after each iteration the values of 2e,

1ϕ ϕ α= − , 2ϕ ϕ= , and 3ϕ ϕ β= + (and since α and β are known, only one of the angles,

in our case φ, is an independent computed output). The results for Mars are:

Exact Computation (Ptolemy’s Result)

Mars 2e φ1 φ2 φ3

iteration 1 13;02,21 (13;07) 36;22,18 (36;31)° 45;21,41 (45;13)° 39;10,18 (39;19)°

iteration 2 11;51,17 (11;50) 42;38,40 (42;45)° 39;05,19 (38;59)° 45;26,40 (45;33)°

iteration 3 12;01,23 (12;00) 41;22,20 (41;33)° 40;21,39 (40;11)° 44;10,20 (44;21)°

iteration 4 11;59,48 41;34,03° 40;09,56° 44;22,03°

iteration 5 12;00,03 41;32,14° 40;11,45° 44;20,14°

iteration 6 12;00,00 41;32,31° 40;11,28° 44;20,31°

Ptolemy proceeds to check his results by using 2e = 12;00 and his values for φi to

compute the distance from the apsidal line for each opposition. He gets 34;30°(34;30,00),

33;20°(33;20,01), and 52;56°(52;56,07) (the accurately computed values, using his values

for 2e and φ, are given in parentheses). Combining these in pairs returns the starting

values for γ and δ, accurate to the nearest arcmin: 34;30° + 33;20° = 67;50°, and 180° –

(33;20° + 52;56°) = 93;44°. Since the longitude of the third opposition was given by

Ptolemy as 242;34°, he concludes that the longitude of apogee is


, 242;34 52;56 180 115;30A = + + =

which indeed agrees with the result of exact computation.

In order to complete the determination of the model parameters, Ptolemy produces two

additional observations. The first (number four in the table above) is used in combination

with the third opposition to determine the epicycle radius r. The second (number five in

the table above) is derived from ancient records and is used, according to Ptolemy, to

determine an accurate value for Mars’ mean motion in anomaly. Ptolemy reports that the

epicycle radius is 39;30 while accurate computation using Ptolemy’s input data gives

39;34,54. The discrepancy is due to rounding error.9

Ptolemy then reports that at the moment of the fifth observation the distance of Mars

from the apogee of its epicycle was 109;42°, while accurate computation again using

Ptolemy’s input data gives 109;45,10°. This particular discrepancy is quite significant in

its context, since Ptolemy is using the number to determine the mean motion in anomaly

to six sexagesimal places of accuracy. The value he quotes in Almagest 9.1 is

0;27,42,40,19,20,58 °/d . Using Ptolemy’s value for the ancient longitude and Ptolemy’s

time interval, one finds that the final digits of the mean motion are 19,28,7. If you add or

subtract just 1´ to Ptolemy’s longitude, the finals digits are 20,54,35 and 18,1,39,

respectively. Thus it is likely that Ptolemy was simply choosing the best value to 1´


precision, among the neighboring values, that combined with his value of the anomaly at

time t3, results in a value of ωa closest to the value he quotes in Almagest 9.1.

Having now fully determined the equant model parameters, Ptolemy extrapolates over the

475 years from the fifth (and already most ancient) observation back to his chosen

Nabonassar epoch to determine the model epoch values, which he quotes as 3;32° in

longitude, 327;13° in anomaly, and 106;40° for longitude of apogee. Accurate

computation gives 3;30,03° for longitude, 327;16,41° for anomaly, and 106;39,52° for

longitude of apogee. In some sense, these small ‘adjustments’ being made by Ptolemy are

required, since he would certainly want the sum of the epoch values in mean longitude

and anomaly to equal the mean longitude of the Sun at the same moment, which he takes

as 330;45°. Altogether, though, these relatively minor discrepancies lead to small

differences between the input data and the model output for the same moments in time.


Jupiter

The input data for Jupiter are:

Ptolemy Model

t1 1769773.45830

t2 1770975.41830

t3 1771377.70830

t4 1772018.70830

t5 1633644.74997

λ1 233;11° 233;10,45°

λ2 337;54° 337;53,05°

λ3 14;23° 14;22,22°

λ4 75;45° 75;42,12°

λ5 97;33° 97;30,52°

λ1(S) 53;11,12°

λ2(S) 157;52,51°

λ3(S) 194;23,32°

As in the case for Mars, the third column gives the longitudes predicted for Jupiter and

the mean Sun using the final Almagest models. Ptolemy’s values for α and β compare to

exactly computed values, using Ptolemy’s times and longitudes, as follows:


Ptolemy Exact

α 99;55° 99;54,35°

β 33;26° 33;26,21°

γ 104;43° 104;43°

δ 36;29° 36;29°

The results for Jupiter’s iterations are:


Jupiter 2e φ1 φ2 φ3

iteration 1 5;20,03 (5;23)° 79;00,24 (79;30)° 1;04,35 (0;35)° 32;21,24 (32;51)°

iteration 2 5;28,55 (5;30)° 77;34,40 (77;15)° 2;30,19 (2;50)° 30;55,40 (30;36)°

iteration 3 5;29,40° 77;22,51° 2;42,08° 30;43,51°

iteration 4 5;29,44° 77;21,28° 2;43,31° 30;42,28°

iteration 5 5;29,44° 77;21,19° 2;43,40° 30;42,19°

Ptolemy again checks his results by using 2e = 5;30 and his values for φi to compute the

distance from the apsidal line for each opposition. He gets 72;11°(72;11,12), 3;06°

(3;06,19), and 33;23°(33;22,55) (the accurately computed values are given in

parentheses). When combined these again return precisely the intervals for γ and δ: 180°

– (72;11° + 3;06°) = 104;43° and 3;06° + 33;23° = 36;29°. Since the longitude of the


third opposition was given by Ptolemy as 14;23, he concludes that the longitude of

apogee is

. 14;23 33;23 180 161;00A = − + =

Accurate computation, however, gives 160;53°, a discrepancy that must occur since

Ptolemy’s final values for the φi do not agree with the results of exact computation.

As he did for Mars, Ptolemy produces two additional observations to determine the

epicycle radius r and an accurate value for Jupiter’s mean motion in anomaly. Ptolemy

reports that the epicycle radius is 11;30 while accurate computation using Ptolemy’s

input data gives 11;35,45. Ptolemy then reports that at the moment of the fifth

observation the distance of Jupiter from the apogee of its epicycle was 77;02°, while

accurate computation again using Ptolemy’s input data gives 77;04,44°.


nearly 507 years from the fifth (and already most ancient) observation back to his chosen


longitude, 146;04° in anomaly, and 152;09° for longitude of apogee. Accurate

computation gives 184;38,19° for longitude, 146;06,40° for anomaly, and 152;8,52° for

longitude of apogee. Ptolemy was probably again adjusting values for consistency with

his solar theory. Just as for Mars, these relatively minor discrepancies thus lead to small

differences between the input data and the model output for the same moments in time.


Saturn

The input data for Saturn are:

Ptolemy Model

t1 1767529.25000

t2 1769790.16667

t3 1770921.00000

t4 1771818.33333

t5 1637841.25000

λ1 181;13° 181;13,00°

λ2 249;40° 249;39,05°

λ3 284;14° 284;14,08°

λ4 309;10° 309;05,13°

λ5 159;30° 159;27,18°

λ1(S) 1;12,57°

λ2(S) 69;39,18°

λ3(S) 104;14,39°

Once again, the third column gives the longitudes predicted for Saturn and the mean Sun

using the final Almagest models. Ptolemy’s values for α and β compare to exactly

computed values, using Ptolemy’s times and longitudes, as follows:


Ptolemy Exact

α 75;43° 75;42,53°

β 37;52° 37;52,11°

γ 68;27° 68;27°

δ 34;34° 34;34°

The results for Saturn’s iterations are:


Saturn 2e φ1 φ2 φ3

iteration 1 7;03,32 (7;08) 56;16,34 (57;43)° 19;26,25 (19;51)° 57;18,25 (55;52)°

iteration 2 6;48,59 (6;50) 57;42,45 (57;05)° 18;00,14 (18;38)° 55;52,14 (56;30)°

iteration 3 6;49,55 57;20,36° 18;22,23° 56;14,23°

iteration 4 6;49,51 57;25,15° 18;17,44° 56;09,44°

iteration 5 6;49,51 57;24,19° 18;18,40° 56;10,40°

Once again Ptolemy checks his results by using 2e = 6;50 and his values for φi to

compute the distance from the apsidal line for each opposition. He gets 51;47°(51;46,45),

16;40°(16;39,20), and 51;14°(51;14,00) (the accurately computed values are given in

parentheses). And once again he recovers precisely the values for γ and δ: 51;47° +

16;40° = 68;27° and 51;14° – 16;40° = 34;34°. Since the longitude of the third opposition

was given by Ptolemy as 284;14°, he concludes that the longitude of apogee is


. 284;14 51;14 233;00A = − =

And in this case, accurate computation gives 233;18°, again reflecting the fact that

Ptolemy’s final values for the φi do not agree with the results of exact computation.

As he did for Mars and Jupiter, Ptolemy produces two additional observations to

determine the epicycle radius r and an accurate value for Saturn’s mean motion in

anomaly. Ptolemy reports that the epicycle radius is 6;30 while accurate computation

using Ptolemy’s input data gives 6;31,41. Ptolemy then reports that at the moment of the

fifth observation the distance of Saturn from the apogee of its epicycle was 183;17°,

while accurate computation again using Ptolemy’s input data gives 183;15,04°.


518 years from the fifth (and already most ancient) observation back to his chosen


mean longitude, 34;02° in anomaly, and 284;10° for longitude of apogee. Accurate

computation gives 296;45,21° for mean longitude, 33;59,38° for anomaly, and

284;08,58° for longitude of apogee, and as before probably adjusted for consistency. Just

as before, these relatively minor discrepancies thus lead to small differences between the

input data and the model output for the same moments in time.


Comments

The attentive reader might have noticed a few instances where Ptolemy’s account is not

consistent with the results of exact calculation. First, for Mars Ptolemy’s values for 2e

and φi after three error-prone iterations agree virtually exactly with the values reached

after accurate computation to full convergence. Such a circumstance is not impossible,

but seems unlikely.

Second, for Jupiter and Saturn Ptolemy’s values after two equally error-prone iterations

are closer to the accurately computed and fully converged values than to the values

Ptolemy would have gotten by computing accurately and stopping at two iterations.

Nevertheless, the agreement is not as good as we noticed for Mars.

Finally, Ptolemy’s values for 2e and A are nicely round values for all three planets. The

likelihood of that happening from purely empirical data cannot be very large. One might

suspect simply convenient rounding, but that cannot be the case for the apogee values,

which are accurately computed as round numbers with no rounding involved.

So how might we reconcile our reconstruction of the correct calculations with Ptolemy’s

own quite specific account? One option would be to accept his claims that he determined

the times and longitudes of the oppositions from empirical observations, but then assume

that he simply followed the iterative algorithm far enough to see where it was headed,

and finally steered it to round numbers for 2e and A.


However, reality cannot be this simple. In this scenario the correlation between Ptolemy’s

input data and his final rounded numbers would be effectively washed out, so that those

numbers could never reproduce the original data. Yet Ptolemy’s verification for all three

planets indeed does combine his round value for 2e and his computed values for the φi,

each of which is in error by anywhere from 10´ to 40´ relative to what he would have

gotten by computing accurately for two or three iterations, and he invariably recovers,

accurate to an arcmin, his input values for γ and δ. Such accurate correlation between

input data and poorly computed, rounded output parameters is impossible, yet it occurs

for all three planets.10

An alternative, and almost as direct, explanation is that Ptolemy did exactly what we did:

start with purported observations and perform many more iterations than he told us about.

Then he would have discovered the correct final values, and he could go back and steer

his calculations of the first few iterations to arrive at what he knew to be the correct

values for 2e and φi. This cannot be the case, either, but to understand why requires some

detailed discussion.

We first must consider whether the cases of Jupiter and Saturn can be construed as

evidence in favor of the idea that Ptolemy’s data is, in fact, derived from empirical

observation. The answer to that question is no, and the proof is implicit in Ptolemy’s own

calculations.


Let us consider Saturn first. After completing two iterations Ptolemy announces that his

final values are 2e = 6;50, φ1 = 57;05°, φ2 = 18;38°, and φ3 = 56;30°. He then computes

the value of ψi implied by each pair (2e, φi). The telling point is that for the second pair

his purported verification actually fails. He quotes ψ2 = 16;40°, while accurate

computation gives 16;39,20°, which of course rounds accurately to 16;39°. Such a trifling

mistake would, in most circumstances, be of no interest, but in the present context

Ptolemy is consistently verifying to 1´ accuracy, and so we have to be careful.

Combining the ψi’s now gives γ = 68;26° and δ = 34;35°, instead of 68;27° and 34;34°.

Using the quartet (α, β, γ, δ) = ( 75;43°, 37;52°, 68;26°, 34;35°) and iterating to

convergence produces 2e = 6;49,50, φ1 = 56;58,22°, φ2 = 18;44,37° and φ3 = 56;36, 47°,

all values now quite close to Ptolemy’s final values. Computing ψ3 now gives 51;20,19°,

and combining with λ3 gives A = 232;53,41°, again much closer to Ptolemy’s 233;00°.

Thus it is quite likely that somewhere along the line Ptolemy made a small slip in

estimating the circumstances of the second opposition.

The case of Jupiter is similar. During his verification Ptolemy finds φ1 = 72;11° and φ2 =

3;06°, which combine to γ = 180° – (φ1 + φ2) = 104;43°. However, exact calculation gives

γ = 180° – (72;11,12° + 3;06,19°) = 104;42,29°, which rounds to 104;42°. Using the

quartet (α, β, γ, δ) = ( 95;55°, 33;26°, 104;42°, 36;29°) and iterating to convergence

produces 2e = 5;29,33, φ1 = 77;14,15° φ2 = 2;50,44° and φ3 = 30;35, 15°, all virtually

identical to Ptolemy’s final values. Computing ψ3 now gives 33;21,53°, and combining

with λ3 gives A = 161;01,07°, again almost identical to Ptolemy’s 161;00°. Thus it is


again quite likely that somewhere along the line Ptolemy made a small slip, this time

even smaller than for Saturn, and in this case in estimating the circumstances of the first

opposition.

Note that the correcting of the small errors in Ptolemy’s verifications of Jupiter and

Saturn is not the result of some wide search for better numbers, but results entirely from

numbers that Ptolemy provides most directly. Hence, while it might be imprudent to

claim that the cases of Jupiter and Saturn give airtight proofs that Ptolemy was using

other than empirical data for his inputs, it may certainly be claimed that the cases of

Jupiter and Saturn provide no evidence whatsoever in support of any claim that Ptolemy

was using empirical observations as the source of his input data.

Thus the issue is no longer whether Ptolemy started with a set of empirical observations,

analyzed them far enough to see where things were headed, and then steered his

computations to pleasingly round numbers. The issue is rather that Ptolemy started with a

set of input data that indeed converges to precisely the round numbers that Ptolemy

assumes as his final values. Such a numerical accident is very unlikely for even one

planet, and of course completely out of the question for all three. The three cases of

computing apogee, where combining two very non-round numbers always produces a

round one, is the most compelling argument against Ptolemy’s claim that he really relies

on empirical input data.


All of this strongly suggests that the times and longitudes of opposition that Ptolemy

reports in the Almagest are not, in fact, the result of data reduction based on observation.

Perhaps the simplest scenario consistent with our analysis is that Ptolemy did indeed

begin with some estimates of time and longitude based on observation, and that he did

indeed proceed to analyze these with the iterative algorithm far enough to see where

things were headed. He then chose round values for 2e and A, and finally used these and

the equant model to recompute what his input data needed to be to produce the final

round values he had chosen, and it is this tweaked data that we find in the Almagest.

An equally tenable alternative scenario is that Ptolemy dispensed with observation

altogether and simply used the equant model with a priori known, fairly round

parameters to compute the circumstances of each opposition. This would not be hard to

do, since the approximate dates of mean opposition were certainly known a priori. One

would simply compute longitudes of the planet and the mean Sun a few, perhaps 10, days

before and after the expected opposition, and use these to estimate the date of opposition

by interpolation. The estimate would certainly be close, but suppose it was slightly early.

One would then compute longitudes of the planet and the mean Sun a short time later, but

after opposition, and use this new pair to re-estimate opposition. Such a process

obviously converges quickly and accurately and requires minimal effort.

It is equally simple to imagine how the round values for 2e and A arose in the first

place.11 If one has a sequence of empirical observations, it is straightforward to look at

the difference in longitude of consecutive oppositions and observe that the longitude


differences indeed vary as the pair moves around the ecliptic, suggesting that the

longitude of minimum pair-wise difference is the direction of apogee for the planet.

Having determined apogee, it is straightforward to find the eccentricity and the radius of

the epicycle. Or, with somewhat more work, one could just do a trio analysis on a dozen

or so trios of oppositions and estimate the most likely values of the parameters, perhaps

by taking the median. Under any such scenario, we would certainly expect nicely round

values for the parameters, just as Ptolemy quotes.

In the end it is clear that Ptolemy’s manipulation of data for the outer planets is

completely consistent with his treatment of the inner planets, the Sun and the Moon. In

each case he produces purported ‘observations’ which he claims he made, and which he

claims are the empirical basis for his model parameters. And in each case those claims

have been found to be not the case. Especially in the case of the outer planets, the original

source of the model parameters must be considered as unknown.


Appendix A

1. Using three oppositions to find the eccentricity and apsidal direction for the bisected

equant model

Ptolemy’s method for solving this problem is based on classical geometric analysis, and

generally works in terms of angles in the range 0° to 90, so that the algebraic signs of

terms are never negative. This requires that each configuration of input angles be

analyzed by inspection. The solution presented below is mathematically, but not step-by-

step, equivalent, and allows the angles to have any values and any combination of

algebraic signs. It is thus far more convenient for computer calculation.12

First consider an eccentric deferent model. We have a deferent circle of unit radius

centered at point F. Let point E be the position of the Earth, so that the apogee A lies on

the extension of the line EF. The length of EF is 2e (in anticipation of the equant). Let λ1,

λ2 and λ3 be three longitudes of the planet at opposition, and let points Q1, Q2 and Q3 be

the center of the planet’s epicycle at the times t1, t2 and t3 of the oppositions, so that the

lines FQi, i = 1,3 are all (unit) radii of the deferent circle. Let angle AFQ2 = φ, angle

AEQ2 = ψ, angle FQ2E = ζ = φ - ψ , angle Q1FQ2 = α, angle Q2FQ3 = β, angle Q1EQ2 = γ,

and angle Q2EQ3 = δ.

Given the mean speed ω of the planet’s epicycle center on the deferent and the times of

the oppositions, the angles α and β are determined by 2 1(t t )α ω= − and 3 2( )t tβ ω= − .


The angles γ and δ are determined by 2 1γ λ λ= − and 3 2δ λ λ= − . We have ϕ ζ ψ= + , so

the problem is to determine e, ψ, and ζ in terms of α, β, γ, and δ.

Applying the law of sines to triangle EFQ2 gives 2 sin sine ψ ζ= . Similarly triangle

EFQ1 gives 2 sin( ) sin( )e ψ γ ζ α− = − + γ and triangle EFQ3 gives

2 sin( ) sin( )e ψ δ ζ β+ = + −δ . Expanding these latter two equations gives

2 sin cos 2 cos sin sin cos( ) cos sin( )e eψ γ ψ γ ζ α γ ζ α− = − − γ−

and

2 sin cos 2 cos sin sin cos( ) cos sin( )e eψ δ ψ δ ζ β δ ζ β+ = − + δ− ,

and letting 2 cosx e ψ= , sin 2 siny eζ ψ= = , and cosz ζ= , we get

cos sin cos( ) sin( )y x y zγ γ α γ α− = − − − γ

and

cos sin cos( ) sin( )y x y zδ δ β δ β+ = − + −δ .

From these we get

sin sin( ) sin sin( )tansin( ) cos sin( ) cos sin( )

yx

δ α γ γ β δψα β γ δ γ β δ δ α γ

− − −= =

+ − − − − − −


sin sin( ) sin sin( )tansin cos( ) sin cos( ) sin( )

yz

δ α γ γ β δζγ β δ δ α γ γ δ

− − −= =

− + − − +

The signs in the equations for tanζ and tanψ are such that if the denominator in the

equation for tanζ is positive then ζ and ψ will be in the correct quadrant (e.g. when using

the atan2(y,x) function in some programming languages). If that denominator is not

positive, then reverse the signs in the numerators and denominators of both equations.

E

A

F

Q1

Q2

Q3

αβγδ

Finally,

ϕ ζ ψ= + and sin2sin

e ζψ

=

and the longitude of the apogee at time t2 is λA = λ2 – ψ.


In order to correct for the fact that the equant point F is not really at the center of the

deferent, an iterative algorithm is used.

φ

ψε

E

M

F

ZQ

A

Consider a new point M a distance e from E in the direction of point A, hence bisecting

EF, and let M be the center of a (second) deferent circle of unit radius. The planet’s

epicycle will be centered at point Z on this new circle. The extension of the line FZ

intersects the original circle at point Q. Since motion is still considered uniform about the

point F, we still have angle AFZ = φ and now angle AMZ = χ. Angle AEZ is still ψ, so

we have angle FZM = φ – χ and angle MZE = χ – ψ. Let angle ZEQ = ε and let angle

MEQ = ψ ′ . Then letting Z be the three planetary positions in turn, we proceed as

follows, with the various subscripts denoting the three planetary positions: let ψ1 = ψ – γ,


ψ2 = ψ, ψ3 = ψ – δ, and let φ1 = φ – α, φ2 = φ, and φ3 = φ – β. Then

1sin ( sin )i i ieχ φ −= − φ , i = 1,3, and we estimate εi from

sin sin( )tancos cos( )

i ii

i i

ee

ψ χ ψεψ χ ψ

− −=

+ −,

which we use to find 2 2ψ ψ ε′ = + . We then ‘correct’ the input values of γ and δ according

to

2 1new original ( )γ γ ε ε= + −

and

3 2new original ( )δ δ ε ε= + − .

The iteration continues by restarting the algorithm described in the first part of this

section with the new values of γ and δ (note that the original values never change), and

continues until the values of ε1, ε2, and ε3, and hence e and λA, stabilize.

Appendix B

In some cases the trio algorithm discussed in Appendix A can be extremely sensitive to

small variations in the values of the input variables α, β, γ, and δ. Ptolemy is certainly

generally aware of the numerical sensitivity in these trio analyses, and in fact uses it in

Almagest 4.11 to explain how relatively small errors in the computed intervals of time

and longitude led Hipparchus astray while analyzing two lunar trios.


The tables below show the changes in the second sexagesimal place of the output

variables φ3, ψ3, 2e, and A (thus in units of arcseconds for the angles) that result from a

change from Ptolemy’s values of one unit in the first sexagesimal place (thus 1´) of the

input angles α, β, γ, and δ:

Mars ψ3 φ3 2e A

α 0,08 0,39 0,47 -0,08

β 3,22 2,57 0,01 3,20

γ 0,10 0,46 -0,56 0,10

δ 3,57 3,28 -0,01 -3,57

Jupiter ψ3 φ3 2e A

α 7,31 6,52 -0,10 7,21

β 13,18 12,21 -1,34 -12,32

γ 7,40 7,00 0,12 -7,42

δ 10,35 10,30 1,30 11,30

Saturn ψ3 φ3 2e A

α 6,50 7,50 0,40 -8,45

β 15,05 15,45 0,35 15,00

γ 7,15 8,15 -0,40 7,15

δ 17,14 17,50 -0,40 -16,10


Thus, for example for Jupiter, a change in γ of 1´ from 104;43° to 104;42° (as in fact

happens at one point in the discussion in the main text) changes the longitude of apogee

by about 7⅔´. For Saturn, if one decreases γ by 1´ and increases δ by 1´(as also happens

in the discussion above), then the longitude of apogee changes by about 23½´. This

extreme level of sensitivity for Jupiter and Saturn, compared to Mars, is the result of two

factors. First, the size of the eccentricity for Mars is about twice as large as for Jupiter or

Saturn, and it is relatively easier to determine the direction of a long line segment than of

a short line segment. Second, for Mars the oppositions are strategically placed near

octants, while for Jupiter and Saturn the second opposition is too close to the direction of

the apsidal line.13

This numerical sensitivity has consequences for Ptolemy’s entire program of using just

five data points to determine the parameters of each outer planet. This can be illustrated

by the following sequence of computational exercises.

First we use the Almagest models for Mars and the mean Sun to compute very accurately

the time and longitudes of the oppositions that Ptolemy used in years 130, 135, and 139.

We also compute the position of Mars at two other times: 3d later and 410y234d earlier

than the moment of the third opposition, corresponding roughly to Ptolemy’s time

intervals for his fourth and fifth observations. When this data is analyzed according to

Ptolemy’s protocol, one gets the values 2e = 11;59,39, r = 39;30,00, and ωa =


0;27,41,40,19,25,50°/d, and the epoch values for mean longitude, anomaly and apogee are

3;31,40°, 327;13,20°, and 106;38,33°, resp. When these values are used to recompute the

longitudes of Mars at the five times used, the comparison is as follows:

time input recomputed

1768888.541666 80;58,54° 80;57, 8°

1770418.326979 148;47,31° 148;46,41°

1771974.420828 242;31,59° 242;32,12°

1771977.420828 241;34, 0° 241;34,13°

1622093.420828 212;39,26° 212;39,26°

The agreement in longitude for the fifth observation is exact because Ptolemy’s protocol

uses the extrapolation from the fifth observation to find his epoch values. The small

disagreements for the first four observations result from Ptolemy’s neglect of the slow

movement of the apsidal line at that stage of his calculations. Note that even with this

degree of care, the errors in the first and second longitudes are 1-2´, and so would affect

Ptolemy’s calculations, which are done to a precision of at least 1´.

As a second exercise, we can use as input the times and longitudes reported by Ptolemy

for each of the five observations, and proceed as above to execute his protocol without

calculation errors. One gets the values 2e = 11;59,34, r = 39;34,53, and ωa =

0;27,41,40,09,28,16°/d, and the epoch values for mean longitude, anomaly and apogee are


3;24,06°, 327;20,54°, and 106;36,52°, resp. When these values are used to recompute the

longitudes of Mars at the five times used, the comparison is as follows:


1768888.54167 81;00° 80;39,50°

1770418.37500 148;50° 148;28,33°

1771974.41667 242;34° 242;04,14°

1771977.35903 241;36° 241;06,08°

1622092.75000 212;15° 212;15,00°

Once again the agreement with the fifth observation is guaranteed, but the errors in the

recomputed values of the first four observations are now much more substantial, typically

30´ or so.

As a final exercise we can use data from real oppositions of Mars with the real mean

Sun.14 Because of Ptolemy’s pervasive systematic error one finds longitudes about 1°

larger than before, but the times are still roughly the same. One gets the values 2e =

11;39,10 , r = 39;26,15, and ωa = 0;27,41,40,16,44,18°/d, and the epoch values for mean

longitude, anomaly and apogee are 3;33,15°, 327;11,45°, and 111;13,16°, resp. When

these values are used to recompute the longitudes of Mars at the five times used, the

comparison is as follows:



1768888.11938 81;43,27° 82;19, 8°

1770418.62428 150;15,41° 150;53,42°

1771973.90440 242;31,59° 241;42,31°

1771976.90440 241;34, 0° 240;44,25°

1622092.90440 212;42, 0° 212;42, 0°

As above the agreement with the fifth observation is guaranteed, but now the errors in the

recomputed values of the first four observations are even larger, typically about 40´.

From these examples we see that if you use input data that are known to be consistent

with the theoretical model in question (in this case the Almagest equant model), then

Ptolemy’s analysis protocol is nearly self-consistent (and could be made completely so by

adding a moving apsidal line into the analysis). But if one uses any other kind of data,

then one will in general not find self-consistent results.

In order to understand why this happening, it is useful to do one last computational

exercise. Let us find all real oppositions of Mars with the real mean sun over some

substantial time frame, e.g. the interval 0 AD – 150 AD. For each such opposition we

also generate two additional real observations, the first 3 days later, the second 410y234d

earlier. We then form successive trios of alternate oppositions, e.g. nos. 1, 3, 5, nos. 3, 5,

7, etc., just as Ptolemy did with his single trio. Each of these trios is analyzed using

Ptolemy’s protocol. The results are shown in Figures 1 and 2. One notices a striking


dependence of the eccentricity and epicycle radius upon longitude. This variation in

effective parameter values is caused by the mismatch between the patterns of speed and

distance variation of the planet in its orbit while following equant motion and Kepler

motion, a crucial point first understood by Kepler himself.15 There is certainly no single

best set of parameters for the equant model, within the variance displayed of about 4% in

2e and 2.5% in r. Thus Ptolemy’s data analysis protocol is unavoidably mixing varying

effective parameters in a way completely beyond his control.


11.4

11.6

11.8

12

12.2

0 60 120 180 240 300 360longitude of second opposition

2e

Figure 1


39

39.2

39.4

39.6

39.8

40


r

Figure 2


5.6

5.8

6.0

6.2

6.4

6.6


e

Figure 3.


References

1 Ptolemy’s Almagest, transl. by G. J. Toomer (London, 1984).

2 J. P. Britton, Models and Precision: the quality of Ptolemy’s observations and

parameters, (Princeton, 1992) and derived from his unpublished 1966 Yale PhD thesis;

C. Wilson, “The Inner Planets and the Keplerian revolution”, Centaurus, 17 (1972) 205-

248; R. R. Newton, The crime of Claudius Ptolemy, (Baltimore, 1977); N. M. Swerdlow,

“Ptolemy’s Theory of the Inferior Planets”, Journal for the history of astronomy, 20

(1989) 29-60.

3 Indeed, except for the Sun there seems no particular reason to assume that Ptolemy even

knew how the various model parameters were originally determined.

4 Technically, Ptolemy determines the longitude of the 4th observation by sighting each

planet both with respect to a star and to the Moon. As always, he gets exactly the same

longitude from both determinations.

5 Newton, op. cit. (Ref. 2) found 2e = 12.00036 by direct numerical solution of the

equant model, thus avoiding any iteration steps.

6 Hugh Thurston, “Ptolemy’s Backwardness”, DIO 4.2 (1994) 58-60.

7 This translation was provided by Alexander Jones.

8 Actually, Ptolemy always gives positive angles referred to either apogee or perigee,

whichever results in an angle less than 90°. In order to facilitate comparison I have

consistently converted my results for φ1, φ2, and φ3 to Ptolemy’s conventions.

9 It is curious that Ptolemy would choose a fourth observation only three days after an

opposition, since the radius of Mars’ epicycle, which advances in anomaly at about 1/2°

per day, will still be pointing nearly straight at the observer. Detailed analysis shows that


if Ptolemy’s observed longitude for the fourth observation changes by 1/4°, his calculated

value for the radius would change from 39;30 to about 39;00.

10 See Appendix B for a detailed discussion of the sensitivity of Ptolemy’s calculations to

changes in his input data.

11 James Evans, The History and Practice of Ancient Astronomy, (New York, 1998), p.

362-368.

12 All the derivations in this appendix follow closely K. Stumpff, Himmelsmechanik, Bd.

1, Berlin, 1956. The original geometrical derivations are clearly explained in O.

Pedersen, A Survey of the Almagest, Odense, 1974; O. Neugebauer, A history of ancient

mathematical astronomy, (3 vols., Berlin, 1975); H. Thurston, Early Astronomy, New

York, 1994.

13 The effect of numerical errors that we are discussing here is several steps removed, and

considerably simpler to discuss, than the effect of errors in the sequence of time and

planetary longitude pairs that must be made to determine the time of opposition. Since

the longitude of opposition is dependent upon the determined time, either directly

through a solar model or less directly through interpolation of measurements, the errors in

time and longitude of opposition will be correlated, and ideally speaking, only the

measured time is a true independent variable. If the correlation is less than 100%, which

will generally be the case, then the analysis is even more complicated.

14 These are computed using programs of Pierre Bretagnon and Jean-Louis Simon,

Planetary Programs and Tables from –4000 to +2800, (Richmond, 1986). The mean Sun

is taken from Jean Meeus, Astronomical Algorithms, (Richmond, 1998) p. 212.


15 Figure 3 shows the result of analyzing a similar set of oppositions from Kepler’s time,

along with the results (large dots) of analyzing the ten oppositions measured by Tycho

Brahe and used by Kepler in Astronomia Nova. The effect we are looking at appears to be

just barely noticeable, remembering that Kepler would not have had the benefit of the

trail of small dots in Fig. 3 to guide his eye. In any event, there is no record of Kepler

doing such an analysis.


Documents

The Almagest Treatment of the Outer Planetsdduke/outer3.pdfinner planets Mercury and Venus, there is a substantial gap between Ptolemy’s description of events and what actually happened.3